Short-Rate Interest Rate Models › Pricing Interest Rate Derivatives

Short-rate models enable pricing of various interest rate derivatives. Let's implement a simple example: pricing a zero-coupon bond option.

Under the Vasicek model, European options on zero-coupon bonds have closed-form solutions. These formulas resemble the famous Black-Scholes formula for stock options, but with modifications appropriate for the interest rate setting. A call option on a $T_2$-maturity bond, expiring at time $T_1 < T_2$, gives the holder the right to purchase the bond at the strike price $K$. The value of this option is:

$$C = P(0, T_2)N(d_1) - K \cdot P(0, T_1)N(d_2)$$

Each term in this pricing formula has a clear interpretation:

• $C$ value of the European call option
• $P(0, T_i)$: price of zero-coupon bond maturing at $T_i$
• $K$: strike price
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1, d_2$: standardized limits for the integration
• $\sigma_P$: volatility of the forward bond price

The first term $P(0, T_2)N(d_1)$ represents the expected present value of receiving the bond, weighted by the probability (under an appropriate measure) that the option will be exercised. The second term $K \cdot P(0, T_1)N(d_2)$ represents the expected present value of paying the strike price, again weighted by the exercise probability under the risk-neutral measure.

The parameters $d_1$ and $d_2$ measure how far the current forward price is from the strike, scaled by volatility:

$$\begin{aligned} d_1 &= \frac{\ln(P(0, T_2) / (K \cdot P(0, T_1)))}{\sigma_P} + \frac{\sigma_P}{2} \\ d_2 &= d_1 - \sigma_P \end{aligned}$$

The numerator of the first term in $d_1$ compares the forward price of the underlying bond (which equals $P(0, T_2)/P(0, T_1)$ in a no-arbitrage setting) to the strike price $K$:

• $d_1, d_2$ standardized integration limits
• $P(0, T_i)$: zero-coupon bond prices
• $K$: strike price
• $\sigma_P$: volatility of the forward bond price

The volatility term $\sigma_P$ measures how much uncertainty there is about the bond price at the option's expiration. In the Vasicek model, this volatility can be computed analytically:

$$\sigma_P = \sigma B(T_1, T_2) \sqrt{\frac{1 - e^{-2\kappa T_1}}{2\kappa}}$$

This expression combines several factors that contribute to bond price uncertainty:

• $\sigma$ volatility of the short rate
• $B(T_1, T_2) = (1 - e^{-\kappa(T_2 - T_1)})/\kappa$: Vasicek bond coefficient
• $T_1$: option expiration time
• $T_2$: bond maturity time
• $\kappa$: mean reversion speed

The factor $\sigma$ is the underlying volatility of the short rate. The factor $B(T_1, T_2)$ translates short-rate volatility into bond price volatility for a bond maturing at $T_2$, as of the option expiration at $T_1$. The square-root term accounts for the accumulation of uncertainty from now until the option expiration, adjusted for mean reversion. Stronger mean reversion (larger $\kappa$) reduces uncertainty because the rate is pulled back toward $\theta$ more quickly.